Lifting Galois representations

Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic ℤp-extensions

“…W * f ) are the distinct elements 1, , and −1 (resp. , 1 and 2 ) of [Ram,§3], these classes are referred to as null cocycles. )…”

Finiteness Conjectures for Fl[[T]]-Analytic Extensions of Number Fields

“…There is numerical and theoretical evidence that there is an abundance of such examples, cf. the probabilistic results in [17].…”

Lifting Galois representations of number fields

“…Apart from being a natural question to ask, the existence of characteristic zero liftings is a fundamental ingredient in the proof Serre's modularity conjecture by Khare and Wintenberger (see [4,5]). The existence of such liftings was first shown by R. Ramakrishna [9]; a modified approach due to Khare and Wintenberger [5], produces liftings with controlled ramification.…”

“…We first generalize the results of Ramakrishna [9], to the setting of number fields following the treatment given in [12]. Secondly, we adapt the method of Khare and Wintenberger [5], and use Skinner and Wiles [11], to give a deformation theoretic approach to the problem of level lowering over totally real fields.…”